DEVELOPMENT OF MATHEMATICAL THOUGHT. 641 



ceptions into algebraical language suggested the inverse 

 operation of interpreting algebraical terms by geometrical 

 conceptions, and led to an enormous extension of geo- 

 metrical knowledge. 1 Further, the infinitesimal methods 

 through which curves and curved surfaces were conceived 

 as being made up of an infinite number of infinitesimally 

 small, straight i.e., measurable lines, led to the in- 

 verse problem ; given any algebraical operations which 

 obtain only in infinitesimally small dimensions i.e., at 

 the limit how do they sum up to finite quantities and 



of mathematics, so long as it 

 was only used to prove theorems 

 which besides being already known, 

 were sufficiently though merely 

 empirically proved. It was H. 

 Grassmann who took up this idea 

 for the first time in a truly 

 philosophical spirit and treated 

 it from a comprehensive point 

 of view." Hankel also refers 

 to Peacock as well as to De 

 Morgan, whose writings, however, 

 he was insufficiently acquainted 

 with (ibid., p. 15). In quite 

 recent times Mr A. N. Whitehead 

 has conceived " mathematics in 

 the widest signification to be the 

 development of all types of formal, 

 necessary, deductive reasoning," 

 and has given a first instalment 

 of this development in his ' Treatise 

 on Universal Algebra' (vol. i., 

 Cambridge, 1 898). See the preface 

 to this work (pp. 6, 7). 



1 A good example of the use of 

 the alternating employment of the 

 intuitive (inductive) and the log- 

 ical (deductive) methods is to be 

 found in the modern doctrine of 

 curves. The invention of Descartes, 

 by which a curve was represented 

 by an equation, led to the intro- 

 duction of the conception of the 

 " degree " or " order " of a curve 

 and its geometrical equivalent ; 



VOL. n. 



whereas the geometrical concep- 

 tion of the tangent to a curve led 

 to the distinction of curves ac- 

 cording to their "class," which 

 was not immediately evident from 

 the equation of the curve but 

 which led to other analytical 

 methods of representation where 

 the tangential properties of curves 

 became more evident. A third 

 method of studying curves was 

 introduced by Pliicker (1832), who 

 started from " the singularities " 

 which curves present, defined 

 them, and established his well- 

 known equations. A further study 

 of these " singularities " led to the 

 notion of the "genus" or "de- 

 ficiency " (Cay ley) of a curve. The 

 gradual development of these and 

 further ideas relating to curves is 

 concisely given in an article by 

 Cayley on " Curve " in the 6th vol. 

 of the ' Encyclopaedia Britannica,' 

 reprinted in Cayley's collected 

 papers, vol. xi. This article fur- 

 nishes also a good example of the 

 historical treatment of a purely 

 mathematical subject by showing, 

 not so much the progress of mathe- 

 matical knowledge of special things, 

 as the development of the manner 

 in which such things are looked at 

 i.e., of mathematical thought. 



2 s 



